As a consequence, there always exists a (Borel) mapping \(t:{\mathcal {M}}\rightarrow {\mathcal {M}}\)  such that \(t_{\#}\mu = \nu \)  where \(t\)  is of the form of e:ot. The issue of regularity and smoothness of OT maps is a delicate one and has been extensively studied (see, e.g., [1]}, [2]}, [3]}); in general, OT maps are not smooth, but can be seen as a natural generalization to normalizing flows, relaxing the smoothness of \(s\) . Henceforth, we will call OT maps “flows.” In fact, our discrete \(c\) -concave functions, the gradient of which are shown to approximate general OT maps, define piecewise smooth maps.
